Heat 1 tbsp olive oil in a large non-stick skillet over medium heat. Add the diced onion and the diced yellow, green and red bell peppers and sprinkle with 1 tsp paprika. Cook, stirring occasionally, until the vegetables are softened, about 4–5 minutes. Transfer the vegetables to a plate and set aside.
Wipe the skillet if needed, then heat the remaining 1 tbsp olive oil over medium-high heat. Season both sides of the 2 chicken breasts with the fajita spice mix to your taste. Add the chicken to the skillet and cook until browned on the outside and cooked through (no pink in the center), about 5–8 minutes per side depending on thickness. Transfer the chicken to a cutting board, let rest 2–3 minutes, then dice into bite-sized pieces. Set aside.
Reduce the skillet heat to medium. You will use 2 tortillas per quesadilla (8 tortillas = 4 quesadillas). Place one tortilla in the skillet, sprinkle some of the grated cheese over the tortilla, add a portion of the diced chicken and cooked peppers, then sprinkle more cheese on top and cover with a second tortilla. Divide the cheese, chicken and peppers evenly as you make each quesadilla.
Cook until the bottom tortilla is golden brown and the cheese starts to melt, about 2–3 minutes. Carefully flip the quesadilla (use a spatula or slide it onto a plate and invert it back into the pan) and cook the other side until golden and the cheese is fully melted, another 2–3 minutes. If the tortillas are browning too quickly before the cheese melts, reduce the heat to medium-low.
Transfer the cooked quesadilla to a cutting board, let rest 1 minute, then slice into quarters. Repeat steps 3–5 with the remaining tortillas, cheese, chicken and peppers. Serve warm.